The invention relates to a method and a system for realising a correlation, at least in space, of two or more electromagnetic fields at the focus of an optical focusing element.
It is extremely important for manufacturers and users of optical focusing elements, such as lenses and lens systems, to be able quantitatively to determine the properties of a lens. For manufacturers the point at issue here is both the testing of newly designed products and routine control of the quality of products. For the users of lenses and lens systems for various applications, quantitative data are of importance for the interpretation of measurement data. Currently there are various techniques which can be used to determine the quality of lenses. Virtually all of these techniques can be assigned in two main categories: (i) the determination of the optical transfer function (OTF) [K. R. Barnes. The optical transfer function (Adam Hilger, London 1971)]; or (ii) the determination of the point spread function (PSF) [G. J. Brakenhoff, P. Blom and P. Barends. Confocal scanning light microscopy with high aperture immersion lenses. J. Microsc. 117, 1979, pp. 219-232].
The principle of all OTF measurement methods is based on the assumption that a lens system can be considered as a linear filter with regard to two-dimensional spatial frequencies in an object when two conditions are met [M. Born and E. Wolf. Principles of optics (Pergamon Press, Oxford, 1980)]: (i) isoplanatism, i.e. the wave front aberrations as a consequence of the imaging lens are independent of the position of the object; and (ii) linear superposition, i.e. the total effect on the image point which is produced by one or more objects is given by a linear summation of the individual values at the image point of each of the objects. Under these assumptions, the PSF is related to the OTF by a Fourier transformation.
There are two basic principles from which all OTF measurement methods are derived: (i) scanning methods; and (ii) shear interferometry.
The scanning methods are based on the use of a sinusoidal grating for the selection of a Fourier component of the object to be tested. The grating can be used either in the object or in the image. The limitations of the techniques which are based on this principle are, in general, a consequence of the low photon efficiency which has an adverse effect on the accuracy of the measurement. Moreover, it is frequently not easily possible to determine the transfer function at spatial frequency zero (necessary for normalization of the OTF) and at high spatial frequencies. The necessary assumption of isoplanatism also constitutes a limiting factor in the applicability of these techniques.
The techniques which are based on shear interferometry determine the autocorrelation function of the exit aperture plane of the imaging system. It is possible to derive [J. J. Stamnes. Waves in focal regions (IOP Publishing Ltd., Bristol, 1986)] that, in the case of incoherent illumination, the OTF is equal to this autocorrelation function. This also applies, with small modifications, to the case of coherent illumination. Limitations of techniques which are based on shear interferometry are that only the monochromatic OTF can be determined and that, in general, highly complex optical systems are needed.
A limitation of all OTF measurement methods is that they are not capable of determining the response of the system in three dimensions. Such information is, in particular, important for, for example, (confocal) microscopy, which is able to make three-dimensional images with sub-micron resolution.
The point spread function (PSF) also can be determined in various ways. In most cases the methods are based on the--three-dimensional--imaging of a point object with a typical size which is much smaller than the characteristic width of the PSF. The PSF is then either determined in the complete field by means of mathematical reconstruction or is measured directly by scanning through movement of point object or detector. Although these methods are able to determine the PSF with high accuracy, they have the significant disadvantage that they are relatively slow--frequently more than one hour is needed for a single determination--and therefore demand long-term (nanometer) stability of the optical measurement set-up. The slowness of this type of method also makes them impractical for routine alignment or test procedures. Since the optical and electronic measurement set-up for these methods is in general highly complex, it is difficult to make the measurements reproducible. Moreover, these methods frequently demand a specific preparation of the sample, whereby, for example, fluorescent--or scattering--latex beads have to be embedded in a specific medium. The choice of the sample is then limited by the availability of fluorescent beads and the typical refractive index of the embedding medium. Consequently, it is frequently not possible when determining the PSF to simulate the specific experimental conditions, both with respect to the properties of the fluorescent substance and with respect to the refractive index of the solvent.